Orthogonal Polynomials and Painlevé Equations Australian Mathematical Society Lecture Series

There are a number of intriguing connections between Painlevé equations and orthogonal polynomials, and this book is one of the first to provide an introduction to these. Researchers in integrable systems and non-linear equations will find the many explicit examples where Painlevé equations appear in mathematical analysis very useful. Those interested in the asymptotic behavior of orthogonal polynomials will also find the description of Painlevé transcendants and their use for local analysis near certain critical points helpful to their work. Rational solutions and special function solutions of Painlevé equations are worked out in detail, with a survey of recent results and an outline of their close relationship with orthogonal polynomials. Exercises throughout the book help the reader to get to grips with the material. The author is a leading authority on orthogonal polynomials, giving this work a unique perspective on Painlevé equations.

1. Introduction; 2. Freud weights and discrete Painlevé I; 3. Discrete Painlevé II; 4. Ladder operators; 5. Other semi-classical orthogonal polynomials; 6. Special solutions of Painlevé equations; 7. Asymptotic behavior of orthogonal polynomials near critical points; Appendix. Solutions to exercises; References; Index.

Walter Van Assche is a professor of mathematics at the Katholieke Universiteit Leuven, Belgium, and presently the Chair of the SIAM Activity Group on Orthogonal Polynomials and Special Functions (OPSF). He is an expert in orthogonal polynomials, special functions, asymptotics, approximation, and recurrence relations.